Filling Riemann surfaces by hyperbolic Schottky manifolds of negative volume

TL;DR

This paper establishes conditions under which a Riemann surface can be realized as the boundary of a convex co-compact hyperbolic 3-manifold with negative volume, using short closed curves on the surface.

Contribution

It provides new criteria involving short hyperbolic curves for filling Riemann surfaces with hyperbolic handlebodies of negative volume.

Findings

Riemann surfaces can be filled by hyperbolic handlebodies under certain length conditions.

Shorter hyperbolic curves on the surface facilitate the construction of the hyperbolic manifold.

The work links geometric properties of curves to the existence of specific hyperbolic 3-manifolds.

Abstract

We provide conditions under which a Riemann surface $X$ is the asymptotic boundary of a convex co-compact hyperbolic manifold, homeomorphic to a handlebody, of negative renormalized volume. We prove that this is the case when there are on $X$ enough closed curves of short enough hyperbolic length.